Question

if p and q are non zero solutions of the equation (x-a)/b + (x-b)/a = b/(x-a) + a/(x-b) then which of the following statement is/are true?i)p and q are of same sign ii)p and q are of opposite sign .justify your answer

Answer 1

(x-a)/b+(x-b)/a=b/(x-a)+a/(x-b)

=>(x-a)/b -b/(x-a)=a/(x-b) - (x-b)/a


if we let (x-a)/b=t
and a/(x-b) = r
now,
t - 1/t = r - 1/r

t - r =1/t - 1/r

(t - r) = -(t - r)/tr

tr = -1 and t =r
now put t and r value
(x-a)/b .a/(x-b)=1

a (x - a)= -b (x-b)
ax -a^2 = b^2 - bx
(a + b) x =(a^2+ b^2)
x=(a^2 + b^2)/(a+b)

again t=r
(x-b)/a =b/(x-a)
(x- a)(x-b)=ab
x^2+(a+b) x +ab=ab
x=0, -(a+b)

here non zero solution -(a+b) and (a^2+ b^2)/(a + b) .

we see both are opposite sign .
option (ii) is correct.

Answer 2

(x-a)/b + (x-b)/a = b/(x-a) - a/(x-b)
--→ (x-a)/b - b/(x-a) = a/(x-b) - (x-b)/a

Suppose (x-a)/b = m and a/(x-b) = n

m-1/m = n-1/n
m-n=1 / m-1/n
(m-n) = - (m-n)/mn
mn= - (m-n)/mn
mn = -1 and m=n

Now putting the value of m and n
(x-a)/b . a/(x-b)=1
a(x-a) = -b (x-b)
ax - a^2 = b^2 - bx
(a+b)x = (a^2 + b^2)
x= (a^2 + b^2)/ (a+b)

Again there will be m=n
(x-b)/a = b/(x-a)
(x-a) (x-b) = ab
x^2 + (a+b) x + ab = ab
x= 0, -(a+b)

I think (ii) option will be correct because the value of x is of the opposite sign.....

Hope u like the answer.
MATK AS BRAINLIEST